Black holes inmodified gravity
Hayato Motohashi（Kogakuin University）
2022.12.16 Online seminar at Waseda UniversityOutline• Introduction• Modified gravity framework• Black holes• SummaryOutline• Introduction• Modified gravity framework• Black holes• SummaryGravity: Last frontier in physics
Newton constant 𝐺 = 6.67430 15 ×10!"" m#kg !"s !$CODATA (2018)

Only four digits of accuracy
SI (2019) definitions:• Extremely weak𝑐 = 299792458 m/s• No negative mass, impossible to shield ℏ = 1.0545718176462×10!"# J s!$%𝑒 = 1.602176634×10

C

Inverse square law and general relativity are verified from
10µm to 1au (∼ 10""m).
Unexplored:• Short range (UV, quantum gravity)• Long range (IR, cosmology)• Dynamical/strong gravity systems
Credit: NASA

Current Universe(1.38×10#$ years)
Late-time acceleration(Dark energy)

Galaxy formation(5×10" years)
Decelerating expansion
First generation star
(3×10" years)
Last scattering(3.8×10! years)
Primordial acceleration(Inflation)

GWstimeCosmic acceleration
GR + SM matter = Decelerating expansion⟹ Requires modification to explain accelerating expansion.
Primordial inflation
Most favored model is “modified gravity”.

Planck 2018

𝑅 + 𝑅! inflation

Starobinsky (1980)

Cosmic acceleration
Late-time acceleration
ΛCDM with tiny cosmological constant,or dark energy/modified gravity?
Planck 2013

𝐻,, 𝜎- tension
also motivates
modified gravity
recently.

Di Valentino, 2011.00246

Asgari+ 2007.15633

Credit: NASA

2016
First
directdetectionof GWs.

LIGO Collaboration

nobelprize.org

GWTC

https://media.ligo.northwestern.edu/gallery/mass-plot
EHT CollaborationLVK observing run plan

https://observing.docs.ligo.org/plan/

EHT Collaboration

NS-NS binary coalescence caused• GW170817• GRB 170817A
Arrival time diff is only ~1.7 sec𝑐%&"#$⟹ −3×10<− 1 < 7×10"#'𝑐

Abbott+ 1710.05832

Constrains modified gravity ~100Hz
Lombriser+, Creminelli+, Sakstein+, Ezquiaga+, Baker+, etc

1.7 sec

Subtle to apply dark energy modelde Rham, Melville 1806.09417

Irrelevant to modified gravity
for primordial Universe andstrong field regime

LIGO Collaboration

2019M87*∼ 16.4 Mpc∼ 6.5×10% 𝑀⊙

BH shadow
EHT Collaboration

Image credit: Kristy Johnson. Photo credits: ALMA -- ESO / C. Malin; SMA - Shelbi Hostler; SPT -- Jeff McMahon; SMT -- David Harvey; PV -- IRAM;
JCMT -- JCMT/JAC; LMT -- Gopal Narayanan; APEX -- APEX.

EHT Collaboration
NASA and The Hubble Heritage Team (STScI/AURA)

Sgr A*∼ 8 kpc∼ 4×10' 𝑀⊙

NASA/JPL-Caltech/ESO/R. Hurt

BHformation

EHT Collaboration
nobelprize.org

Orbit of stars
around BH

GRAVITY Collaboration

Golden age of black holes!

Theoretical framework for testing
gravity is becoming more important.

Outline• Introduction• Modified gravity framework• Black holes• SummaryLovelock theorem
GR is unique theory of gravity under thefollowing assumptions:• Metric theory• 4D diffeo. invariance• 2nd order EL eqs
To go beyond GR we need to consider aviolation of the assumptions.

Lovelock (1971)

Modified gravity

Kaluza (1921) Klein (1926)

Unification in higher dimension• Metric + EM
• 5D diffeo. inv.
• 2nd order EL eqs

Scalar-tensor theory Jordan (1959)• Metric + scalar field• 4D diffeo. inv.
• 2nd order EL eqs

Brans & Dicke (1961)

~2010

Various scalar-tensor theories

𝑓(𝑅) gravity
𝑅 + 𝑅! inflation
𝐾-essence/-inflation
Brans-Dicke
GR (2DOFs)

Conformal transformation
𝑔645 = 𝐴𝑔45 recasts these
theories to scalar-tensor
theories.
Maeda (1988), (1989)

2011~)

𝑋 = 𝜙( 𝜙(𝑆 = 7𝑑 #𝑥 −𝑔 < 𝐿&&'(𝐿( = 𝐺( 𝜙, 𝑋 , 𝐿" = 𝐺" 𝜙, 𝑋 □𝜙
𝐿# = 𝐺# 𝜙, 𝑋 𝑅 − 2𝐺#* 𝜙, 𝑋$

□𝜙 ( − 𝜙+, 𝜙+,𝐿) = 𝐺) 𝜙, 𝑋 𝐺+, 𝜙+, + " 𝐺)* 𝜙, 𝑋

□𝜙 " − 3𝜙𝜙+, 𝜙+, + 𝜙+, 𝜙+- 𝜙-,

Various scalar-tensor theories
are subclasses of Horndeski theory.
Horndeski (1974), Deffayet+ 1103.3260, Kobayashi+ 1105.5723

𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋• Metric + scalar field• 4D diffeo. inv.
• 2nd order EL eqs

Horndeski, “Horndeski Scalar Theory - Past, Present & Future, No.2” (2019)“The text on the frame
describes how I came up withmy scalar-tensor field theory
equations, which are currentlyof interest to many
astrophysicists.
I also describe my simple
theory of quantum gravity,
which is currently only ofinterest to me, and perhaps mycat, who bares no relationshipto Schrödinger’s.”

https://www.horndeskicontemporary.com

Horndeski, “Blue Sky with Black Hole Over Wheat Field” (1993)https://www.horndeskicontemporary.com

~2014

Most general framework?
𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋
Ostrogradsky (1850), Woodard 1506.02210

Higher-order EL eqs mean Ostrogradsky ghosts…?

Degenerate theories

2015~

HM, Noui, Suyama, Yamaguchi, Langlois, 1603.09355
HM, Suyama, Yamaguchi, 1711.08125, 1804.07990
本橋, 物理学会誌 71 (2016) 734「オストログラドスキーの定理：整合的な修正重力理論への道のり」

DHOST

𝐹 𝜙, 𝑋 𝑅𝐴) 𝜙, 𝑋

* GLPV (2014)

𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋∇∇𝜙Langlois+ 1510.06930
Crisostomi+ 1602.03119
Ben Achour+ 1602.08398• Metric + scalar field• 4D diffeo. inv.
• Degenerate higherorder EL eqs

HM, Suyama, 1411.3721, 2001.02483, Aoki, HM, 2001.06756

Non-deg. higher-order theories with Ostrogradsky ghosts

“Cambrian explosion”
Various degenerate theories
were born:• Scalar-tensor

Courtesy: National Science Foundation

Langlois+ 1510.06930, 1512.06820, 1802.03394, Crisostomi+ 1602.03119
Ben Achour+ 1602.08398, 1608.08135, Takahashi+ 1708.02951
De Felice+ 1803.06241, Gao+ 1806.02811, HM+ 2002.07967• Vector-tensor
Allys+ 1609.05870, Kimura+ 1608.07066
Errasti Diez+ 1905.06967, Cadavid+ 2009.03241••••

Boson-Fermion
Generalization of Chern-Simons
Generalization of Fierz-Pauli
Degenerate Hořava

Kimura+ 1704.02717, 1805.10963
Crisostomi+ 1710.04531
Naruko+ 1812.10886
Barausse+ 2101.00641

Quadratic DHOST theories>?
( 𝐴; 𝐿;;<=:𝑆 = # 𝑑 𝑥 −𝑔$$

+ 𝐵@ + 𝐵= □𝜙 + 𝐵? 𝑅$

𝐿" = 𝜙45 𝜙45 , 𝐿$ = □𝜙 $, 𝐿# = □𝜙 𝜙4 𝜙45 𝜙 5 ,𝐿9$

= 𝜙4 𝜙45 𝜙 5: 𝜙: ,𝐿;$=$45
𝜙 𝜙45 𝜙 .

𝐴; (𝜙, 𝑋) and 𝐵; (𝜙, 𝑋) should satisfy degeneracy
condition to avoid the Ostrogradsky ghosts.

Degenerate theories

DHOST

𝐹 𝜙, 𝑋 𝑅∇∇𝜙𝐴) 𝜙, 𝑋
𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋

Non-deg. higher-order theories with Ostrogradsky ghosts

Degenerate theories
Disfomal transformation𝑔̅*+ = 𝐹! 𝜙, 𝑋 𝑔*+ + 𝐹" 𝜙, 𝑋 ∇* 𝜙∇+ 𝜙
Bekenstein, gr-qc/9211017

𝐹 𝜙, 𝑋 𝑅𝐴) 𝜙, 𝑋

“DisformalHorndeski”

𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

DH OST∇∇𝜙No viable cosmology
de Rham, Matas, 1604.08638
Langlois et al., 1703.03797

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋

Non-deg. higher-order theories with Ostrogradsky ghosts

Degenerate theories
Disfomal transformation𝑔̅*+ = 𝐹! 𝜙, 𝑋 𝑔*+ + 𝐹" 𝜙, 𝑋 ∇* 𝜙∇+ 𝜙
Bekenstein, gr-qc/9211017

Langlois, Saito, Yamauchi, Noui, 1711.07403

𝐹 𝜙, 𝑋 𝑅

𝑐<= = 𝑐𝐴) 𝜙, 𝑋

“DisformalHorndeski”

𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

DH OST∇∇𝜙No viable cosmology
de Rham, Matas, 1604.08638Langlois et al., 1703.03797

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋

Non-deg. higher-order theories with Ostrogradsky ghosts

Degenerate theories
Disfomal transformation𝑔̅*+ = 𝐹! 𝜙, 𝑋 𝑔*+ + 𝐹" 𝜙, 𝑋 ∇* 𝜙∇+ 𝜙~2020
Bekenstein, gr-qc/9211017

Langlois, Saito, Yamauchi, Noui, 1711.07403

𝐹 𝜙, 𝑋 𝑅𝑐<= = 𝑐𝐴) 𝜙, 𝑋

“DisformalHorndeski”

𝑓(𝑅) gravity

Horndeski

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Most general framework?

DH OST∇∇𝜙No viable cosmology
de Rham, Matas, 1604.08638
Langlois et al., 1703.03797

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋Non-deg. higher-order theories with Ostrogradsky ghosts

StrategyTheorem: #DOFs is unchanged under invertible transf.
Domenech+ 1507.05390, Takahashi, HM, Suyama, Kobayashi, 1702.01849

Invertible disformal transformation of Horndeski theory
yields DH class, which is more general and viable scalartensor theory.
Disformal transformation𝑔45
̅ = 𝐹,𝑔45 + 𝐹"𝜙4 𝜙5 ,Invertibility condition𝐹, ≠ 0,𝐹, + 𝑋𝐹" ≠ 0,

𝐹> = 𝐹> (𝜙, 𝑋)
𝐹, − 𝑋𝐹,? − 𝑋 $𝐹"? ≠ 0
Bekenstein, gr-qc/9211017

Can we construct more general transformation?

Several attempts• Several types of higher-derivative generalizations wereconsidered, but invertibility condition was not clarified.
Alinea, Kubota, 2005.12747
Minamitsuji, 2104.03662• Invertibility condition on specific metric was studied,which is necessary but not sufficient to identify invertible
transformation.
Babichev, Izumi, Tanahashi, Yamaguchi, 1907.12333, 2109.00912

It was rather used to prove no-go of invertibility:
A transformation𝑔45
̅ = 𝐹,(𝜙, 𝑋)𝑔45 + 𝐹"(𝜙, 𝑋)𝜙4 𝜙5 + 𝐸(𝜙, 𝑋)∇4 ∇5 𝜙with 𝐸 ≠ 0 is non-invertible.

Generalized disformal transformation
By requiring the closedness of the group structure, we
succeeded to derive invertibility (sufficient) condition.
Takahashi, HM, Minamitsuji, 2111.11634

Generalized disformal transformation𝑔45
̅ = 𝐹,𝑔45 + 𝐹"𝜙4 𝜙5 + 2𝐹$𝜙(4 𝑋5) + 𝐹#𝑋4 𝑋5
𝐹> = 𝐹> 𝜙, 𝑋, 𝑌, 𝑍
Invertibility condition
𝐹E ≠ 0,ℱ ≠ 0,

𝑋* ≠ 0,𝑋*F = 𝑋*G = 0,* 𝑍̅𝜕 𝑌,𝜕 𝑌, 𝑍

where

≠ 0,

𝑋 = 𝜙4 𝜙4 ,𝑌 = 𝜙4 𝑋4 ,𝑍 = 𝑋4 𝑋4ℱ = 𝐹,$ + 𝐹, 𝑋𝐹" + 2𝑌𝐹$ + 𝑍𝐹# + (𝑌 $ − 𝑋𝑍)(𝐹$$ − 𝐹"𝐹#)* Further generalization is also possible, as explained in the paper.

Degenerate theories~2020

𝐹 𝜙, 𝑋 𝑅

𝑐<= = 𝑐DH OST∇∇𝜙𝐴) 𝜙, 𝑋
𝑓(𝑅) gravity

Horndeski

No viable cosmology

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋

Non-deg. higher-order theories with Ostrogradsky ghosts𝑐<= = 𝑐
𝑅*+ ∇* 𝑋∇+ 𝑋

GDH“GeneralizedDisformal
Horndeski”

Degenerate theories
2021~
Takahashi, Minamitsuji, HM,
2111.11634, 2209.02176∇∇∇𝜙DH OST𝐹 𝜙, 𝑋 𝑅∇∇𝜙𝐴) 𝜙, 𝑋
𝑓(𝑅) gravity

Horndeski

No viable cosmology

𝑅 + 𝑅! inflation
𝐾-essence/-inflation

𝐺$ (𝜙, 𝑋)

Brans-Dicke
GR (2DOFs)

𝐺( 𝑋

Non-deg. higher-order theories with Ostrogradsky ghosts

Generalized disformal Horndeski (GDH)
Most general scalar-tensor theory at present time>

𝑆DEF [𝑔GH , 𝜙] = # 𝑑 : 𝑥 −𝑔𝐹@ ℱ =/? ( ℒ8 ; [𝑔GH , 𝜙];<=Takahashi, Minamitsuji, HM,2111.11634, 2209.02176

Consistency of matter coupling
When we take into account matter sector, in general,
degeneracy condition is broken by the coupling between
matter and gravity, and Ostrogradsky ghost revives.
DHOST:
No problem for boson and fermion to couple gravity
minimally without invoking ghost.
Deffayet, Garcia-Saenz, 2004.11619

GDH:
Minimally coupled boson introduces extra DOF in general,but there exists a subclass where no extra DOF shows up.
Takahashi, Minamitsuji, HM, 2209.02176
Naruko, Saito, Tanahashi, Yamauchi, 2209.02252

Exploring fermions…
Takahashi, Kimura, HM, in prep.

Outline• Introduction• Modified gravity framework• Black holes• SummaryChandrasekhar, “The Mathematical Theory of Black Holes”

Uniqueness theorem
Rotating BH solution
in GR

⇒

Kerr

only

BHs are characterized by• Mass 𝑀• Spin 𝑎

Vishveshwara (1980)

Uniqueness theorem
Rotating BH solution
in GR

⇒

Kerr

only

How about opposite?
Kerr⇒ Allowed in GR only…?

Vishveshwara (1980)

Uniqueness theorem
Rotating BH solution
in GR

⇒

Kerr

only

How about opposite?
Kerr⇒ Allowed in GR only…? No!

Modified gravity allows various solutions:• Stealth solution(𝑔45 : Kerr, 𝜙: nontrivial profile)• Disformal Kerr solution(𝑔45 : Kerr + deviation, 𝜙: nontrivial profile)• Other solutions* Kerr + trivial scalar can also be a solution for a wide class oftheories.
HM, Minamitsuji, 1804.01731

Vishveshwara (1980)

GRKerr
Photon

EHT Collaboration

GWs
LIGO Collaboration

Modified gravity
Stealth

Disformal Kerr
PhotonPhotonGWs

GWs

Stealth: Background𝑔45 : GR, 𝜙: nontrivial

Various stealth solutions in various theories(Stealth Schwarzschild, stealth Kerr, …)
Babichev+ 1312.3204, Kobayashi+ 1403.4364, Minamitsuji, HM, 1809.06611
Ben Achour+ 1811.05369, HM+ 1901.04658, Charmousis+ 1903.05519

We can identify theories allowing stealth solution bycovariant analysis.
HM, Minamitsuji, 1804.01731, Takahashi, HM, 2004.03883

Stealth: Background𝑔45 : GR, 𝜙: nontrivial

Various stealth solutions in various theories(Stealth Schwarzschild, stealth Kerr, …)
Babichev+ 1312.3204, Kobayashi+ 1403.4364, Minamitsuji, HM, 1809.06611
Ben Achour+ 1811.05369, HM+ 1901.04658, Charmousis+ 1903.05519

We can identify theories allowing stealth solution bycovariant analysis.
HM, Minamitsuji, 1804.01731, Takahashi, HM, 2004.03883
Interesting things are:
• Theories “far” from GR can accommodate stealth sol.
• Nontrivial scalar profile is allowed.
𝑔45 : static or stationary, 𝜙 = 𝑞𝑡 + 𝜓(𝑟)Not like a slightly different theory and/or profile from GR.

Stealth: Perturbation

𝑔45 : GR, 𝜙: nontrivial• Master variables for pert. around static, spherically sym.
spacetime.
(Generalization of Regge, Wheeler (1957), Zerilli (1970) in GR)• Stability condition for theories to allow nonrotating BHsolution.

Kobayashi, HM, Suyama, 1202.4893, 1402.6740
Takahashi, HM, Minamitsuji, 1904.03554, Takahashi, HM, 2106.07128

Stealth: Perturbation

𝑔45 : GR, 𝜙: nontrivial• Master variables for pert. around static, spherically sym.
spacetime.
(Generalization of Regge, Wheeler (1957), Zerilli (1970) in GR)• Stability condition for theories to allow nonrotating BH
solution.

Kobayashi, HM, Suyama, 1202.4893, 1402.6740
Takahashi, HM, Minamitsuji, 1904.03554, Takahashi, HM, 2106.07128

Analysis for BH + Perfect fluid.
Possible application to neutron star.
Kase, Tsujikawa, 2110.12728

For most theories, static scalar field 𝜙 = 𝜙(𝑟)is strongly coupled/unstable.
Minamitsuji, HM, 1809.06611
Minamitsuji, Takahashi, Tsujikawa, 2201.09687
Credit: NASA

Stealth: Perturbation

𝑔45 : GR, 𝜙: nontrivial

Strong coupling/gradient instability with 𝜙 = 𝑞𝑡 + 𝜓(𝑟)de Rham, Zhang 1907.00699

Universal properties for second order theories
0/#（𝜔( = 𝑐.(𝑘 ( ⟹ 𝐸/ ∼ 𝑐. 𝑀 but asymptotically 𝑐.( ≤ 0. 𝑀：cutoff, 𝐸/ : strong coupling scale）

Stealth: Perturbation

𝑔45 : GR, 𝜙: nontrivial

Strong coupling/gradient instability with 𝜙 = 𝑞𝑡 + 𝜓(𝑟)de Rham, Zhang 1907.00699

Universal properties for second order theories
0/#（𝜔( = 𝑐.(𝑘 ( ⟹ 𝐸/ ∼ 𝑐. 𝑀 but asymptotically 𝑐.( ≤ 0. 𝑀：cutoff, 𝐸/ : strong coupling scale）

This is because, when we consider second order theories, weomit higher order terms in

B $
=C

D 9
D E+ C + ⋯ from EFT.
CWith 𝛼 □𝜙 $/𝑀$, we obtain 𝐸F ∼ |𝛼|G/$𝑀
and resolve strong coupling.
（Ostrogradsky ghost show up ∼ 𝑀, below which the theory isdegenerate and ghost-free）

Weak detuning of degeneracy condition ＝ Scordatura Wikipedia
HM, Mukohyama, 1912.00378

Quasinormal modes
Damped oscillations describing ringdown GWs.
LIGO Collaboration

QNMs incomplex 𝜔
plane

Scattering ofGaussian
wave packet
Vishveshwara (1970)

Leaver (1985)

Detection of QNMs and significance of fundamental modes
and overtones are actively discussed recenlty.
Giesler+ 1903.08284, Sago+ 2108.13017, Abbott+ 2112.06861, Cotesta+ 2201.00822
Barausse+ 1404.7149, Cardoso+ 2109.00005, Cheung+ 2111.05415
Berti+ 2205.08547, Kyutoku, HM, Tanaka, 2206.00671

Stealth: QNMs
Usually, we solve the master equation by transforming it
into Schrödinger form or first order form.
For stealth solution, we need a coordinate transformation
Babichev+ 1803.11444
that is different from GR.
1 𝜕$𝜕$− $ $ + $ − 𝑉KLL (𝑟) Ψ = 0𝜕𝑟∗̃𝑐I 𝜕𝑡̃
In general, 𝑡̃ ≠ physical time.
We are interested in evolutionof wave propagation inphysical time.
Dali, The Persistence of Memory (1931)

Stealth: QNMs
S-wave (ℓ = 0) scattering:𝑡̃ = constant surface istimelike.
⟹ Not appropriate for
initial Cauchy surface.

We prepared an appropriate
spacelike hypersurface
and solved the initial value
problem.
Nakashi, Kimura, HM, Takahashi, 2204.05054

Stealth: QNMs
Solved initial value problem of swave (ℓ = 0) with initial Gaussian
wave packet.
• Established the initial value
problem for stealthperturbations.
• S-wave perturbation is
dynamically stable (no growingmode).
• No oscillation. Multipole
perturbations should havesome features.
Nakashi, Kimura, HM, Takahashi, 2204.05054, in prep.

Disformal Kerr

Ben Achour, Liu, HM, Mukohyama, Noui, 2006.07245Anson, Babichev, Charmousis, Hassaine, 2006.06461

Disformal transf. generates solution and identify theory at thesame time.
𝑐23 = 𝑐GDH

Degenerate theories

DH OST𝑔45 : Disformal Kerr

𝑔645 : Stealth Kerr
Horndeski

GR$𝑑𝑠 $ = 𝑑𝑠̃MKNN+ 𝛼 𝑑𝑡 +2𝑀𝑟 𝑟 $ + 𝑎$$𝑑𝑟Δ
Deviation from Kerr is more manifest for rapidly rotating case.

BH shadow
Disformalparameter𝑎/𝑀 = 0(Nonrotating)𝑎/𝑀 = 0.5
(Rotating)𝑎/𝑀 = 0.998(Rapidly rotating)

𝛼 = −0.5

Long+, 2009.07508−0.30 (GR)0.2

Pulsar around BHTakamori+, 2108.13026

SKA

Chaos in particle plunging into BH

Zhou+, 2110.03291

GROutline• Introduction• Modified gravity framework• Black holes• SummaryModified gravity framework
Ø We found invertible generalized disformal transformation,
which enables us to construct GDH theories as the mostgeneral ghost-free scalar-tensor theory.

Black hole solutions
Ø Stealth solution(𝑔45 : Kerr, 𝜙: nontrivial)Ø Disformal Kerr (𝑔45 : Kerr + deviation, 𝜙: nontrivial)

Possible observables
Ø Stealth: We formulated perturbation theory and initial valueproblem. Multipole GWs are expected to have some signals.
Ø Disformal Kerr: Deviations show up in BH shadow, star
motion, pulsar timing etc, which are more manifest for
rapidly rotating BHs.